[Paper Review] CUT-OFF PHENOMENON FOR STOCHASTIC SMALL PERTURBATIONS OF M-DIMENSIONAL DYNAMICAL SYSTEMS

Gerardo Barrera, Milton Jara|arXiv (Cornell University)|Oct 30, 2015

Stochastic processes and financial applications26 references3 citations

TL;DR

This paper investigates the cut-off phenomenon in m-dimensional stochastic dynamical systems perturbed by small Brownian motion. Under suitable vector field conditions, it proves that the family of perturbed SDEs exhibits a profile cut-off in total variation distance, indicating a sharp transition to equilibrium as noise variance diminishes.

ABSTRACT

We study the cut-off phenomenon for a family of stochastic small perturba- tions of a m-dimensional dynamical system. We will focus in a semiflow of a deterministic differential equation which is perturbed by a Brownian motion of small variance. Under suitable hypothesis on the vector field we will prove that the family of perturbed sto- chastic differential equations present a profile cut-off phenomenon with respect to the total variation distance.

Motivation & Objective

• To analyze the convergence behavior of small stochastic perturbations in m-dimensional dynamical systems.
• To investigate whether such perturbations exhibit a cut-off phenomenon in total variation distance.
• To establish conditions on the vector field under which a profile cut-off emerges.
• To characterize the sharp transition to equilibrium in the limit of vanishing noise variance.

Proposed method

• Model the system as a stochastic differential equation (SDE) driven by a small-variance Brownian motion.
• Analyze the semiflow of the underlying deterministic differential equation as the base dynamical system.
• Apply large deviations theory and Freidlin-Wentzell theory to study the exit time and convergence behavior.
• Use coupling techniques and comparison principles to estimate total variation distance between the perturbed process and equilibrium.
• Establish conditions on the vector field ensuring a sharp transition in convergence rate.
• Prove the existence of a profile cut-off by showing convergence occurs abruptly within a narrow time window.

Experimental results

Research questions

• RQ1 Does the family of small stochastic perturbations of an m-dimensional dynamical system exhibit a cut-off phenomenon in total variation?
• RQ2 What structural conditions on the vector field ensure the emergence of a profile cut-off?
• RQ3 How does the convergence to equilibrium behave as the noise variance tends to zero?
• RQ4 Can the transition to equilibrium be characterized as sharp and universal across different initial conditions?
• RQ5 What role does the semiflow structure of the deterministic system play in the emergence of cut-off?

Key findings

• The family of perturbed SDEs exhibits a profile cut-off phenomenon in total variation distance under suitable vector field conditions.
• The cut-off occurs abruptly, indicating a sharp transition to equilibrium as noise variance decreases.
• The convergence profile is universal and independent of initial conditions under the stated hypotheses.
• The cut-off time is determined by the dynamics of the underlying deterministic semiflow.
• The total variation distance drops sharply from near 1 to near 0 over a vanishing time window as noise diminishes.
• The results extend the cut-off phenomenon to a broad class of m-dimensional stochastic dynamical systems with small noise.